FLAT DIMENSION GROWTH FOR C ∗ -ALGEBRAS

By Andrew S. Toms. 2005

51 citations recorded by the source corpus.

Abstract Simple and nuclear C ∗ -algebras which fail to absorb the Jiang–Su algebra tensorially have settled many open questions in the theory of nuclear C ∗ -algebras, but have been little studied in their own right. This is due partly to a dearth of invariants sensitive to differences between such algebras. We present two new real-valued invariants to fill this void: the dimension–rank ratio (for unital AH algebras), and the radius of comparison (for unital and stably finite algebras). We establish their basic properties, show that they have natural connections to ordered K-theory, and prove that the range of the dimension–rank ratio is exhausted by simple algebras (this last result shows the class of simple, nuclear and non- Z -stable C ∗ -algebras to be uncountable). In passing, we establish a theory of moderate dimension growth for AH algebras, the existence of which was first supposed by Blackadar. The minimal instances of both invariants are shown to coincide with the condition of being tracially AF among simple unital AH algebras of real rank zero and stable rank one, whence they may be thought of as generalised measures of dimension growth. We argue that the radius of comparison may be thought of as an abstract version of the dimension–rank ratio.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/72e1032880c878cad5893a12520507a7c449e543

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In Algebra and Number Theory

1. Andrew S. Toms
2. David Kerr
3. N. Christopher Phillips
4. J. Giol
5. W. Winter
6. Huaxin Lin
7. G'abor Szab'o
8. Christopher Phillips

In Geometry and Topology

1. Andrew S. Toms
2. N. Christopher Phillips
3. David Kerr
4. J. Giol
5. W. Winter
6. Hanfeng Li
7. Christopher Phillips
8. Xiangdong Ye

In Mathematical Physics

1. Andrew S. Toms
2. David Kerr
3. N. Christopher Phillips
4. J. Giol
5. W. Winter
6. G'abor Szab'o
7. Huaxin Lin
8. Christopher Phillips

Citations in the corpus

• Comparison Theory and Smooth Minimal C*-Dynamics (2008, background): “A). The radius of comparison of A, denoted by rc(A), is defined to be the infimum of the set {r∈ R+ | Ahas r−strict comparison} whenever this set is nonempty; if the set is empty then we set rc(A) = ∞ ([28]). The condition rc(A) = 0 is equivalent to Ahaving strict comparison (see Subsection 2.2). The ”
• Large subalgebras (2014, background): “1 of [38]) for the equivalence of our formulation with the original version.”
• C*-algebras of stable rank one and their Cuntz semigroups (2018, unknown): “ case (see Section 8 for the relevant definitions): Theorem (8.11, 8.12). Let A be a simple, unital, separable C*-algebra of stable rank one. (i) A has finite radius of comparison in the sense of Toms ([Tom06]) if and only if the subsemigroup W(A) consists precisely of the elements in Cu(A) whose rank”
• Subshifts and perforation (2010, methodology): “We in fact show that one can produce such crossed products with arbitrarily large radius of comparison [27].”